High slew rate input signals come back to summing junction thru negative feedback delayed because of signal path delays. For a moment amplifier has no feedback and overshoot appears at the output (or earlier dependent on design). This will happen to any amplifier if slew rate is not properly limited at the input.The only part of this I disagree with is the phrase "at the input". And yes, these are the very fundamentals of proper frequency compensation. The only nit-pick I would add is if this is indeed done "at the input", that implies a passive network before amplification . . . and while slew-rate and bandwidth are two different things, you can't limit one without the other in a passive network. Which makes it a bandwidth discussion, and again takes us right back to the fundamentals of stability in feedback amplifiers, and phase margin.
Also, a Class AB amp cannot "exhibit higher order of mostly odd harmonics at very low signal levels" as compared to a Class A design . . . very simply because at "very low signal levels", it's a Class A amp. That's its raison d'etre. I could maybe see how other non-crossover nonlinearities (this "kink" you describe) may have been a slight contributor at very specific power levels in days of output devices like 2N3055/2955, but is virtually nonexistant in properly loaded constant-beta modern bipolar power transistors.
But to answer Oem's question . . . I don't think that Class B is "the answer", it's one of many valid options. Why is it not used more? I don't know for sure, but I'd speculate that the main reasons are because it requires extremely accurate thermal bias compensation, and the nonlinear base currents demanded by the drivers from the voltage-amp . . . both present significant (but not insurmountable) design challanges for good results. Class AB is significantly less critical in these regards, but the trade-off is a greater variation in performance with load and signal level, both of which are dynamically working to pull the amp from Class A operation towards Class B. And this transition isn't a particularly graceful one.